2 12 corollary 34 suppose that x ir l is star shaped

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2 12
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Corollary 3.4 Suppose that X IR l is star-shaped with respect to 0 and locally polyhedral at 0 , 0 X and p * IR l . Then pX 0 implies p ( * X ) 0 . Proof. Let P be a polyhedron obtained as an intersection of X with a suffi- ciently small closed cube, whose interior contains 0 . Suppose that there exists ˜ x * X such that p ˜ x < 0 . Then one can find ˜ x 0 = ε ˜ x * P such that p ˜ x 0 < 0 , for some sufficiently small ε * IR ++ . But pP 0 , which by Lemma 3.3 implies p ( * P ) 0 , a contradiction. 2 Proof of Proposition 3.2. Let q p = ( q 1 , . . . , q k ) for some k l. If p = 0 then q 1 = 0 , k = 1 and ¯ B ( p ) = X = B 1 ( p ) . Suppose that there exist a number m { 1 , . . . , k } and an element y X ( q 1 , . . . , q m - 1 ) such that q m y < 0 . Moreover, assume that m is the smallest such a number, which guarantees that q j X ( q 1 , . . . , q j - 1 ) 0 , j ∈ { 1 , . . . , m - 1 } . (7) Note that by construction y B m ( p ) . If m = 1 then ¯ B ( p ) = B ( q 1 ) = B 1 ( p ) by virtue of Proposition 2 . 4 . Assume m ∈ { 2 , . . . , k } and show that ¯ B ( p ) = B m ( p ) . First, we shall prove that ¯ B ( p ) B m ( p ) . Take some arbitrary x ¯ B ( p ) and suppose that x 6∈ B m ( p ) . If so, then by the choice of m and the system of inequalities (7) there exists a number j ∈ { 1 , . . . , m } such that q j x > 0 , q t x = 0 , t = 1 , . . . , j - 1 . (8) Consider some arbitrary non-standard element ˜ x * X such that ˜ x x. Then q j x > 0 implies that q j ˜ x is greater than some strictly positive real number. Take now any standard x 0 X and consider the first non-zero element in the ordered set { q 1 x 0 , . . . , q j - 1 x 0 } . Since j m, it follows again from (7) and the choice of m that such an element, if it exists, is strictly positive. Therefore, a non-standard linear functional λ 1 q 1 + · · · + λ j - 1 q j - 1 takes only positive values on X : λ 1 q 1 + · · · + λ j - 1 q j - 1 · X 0 . Then by Lemma 3.3 λ 1 q 1 + · · · + λ j - 1 q j - 1 · * X 0 . (9) Consider 1 λ j p ˜ x = 1 λ j [ λ 1 q 1 + · · · + λ j - 1 q j - 1 x + q j ˜ x + 1 λ j [ λ j +1 q j +1 + · · · + λ k q k x. 13
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The first component of the sum in the right-hand side is positive (it vanishes if j = 1), the second component exceeds 0 by a non-infinitesimal amount and the third component is infinitesimal. Therefore, (1 j ) p ˜ x is strictly positive, so that p ˜ x > 0 for all ˜ x * X such that ˜ x x. This contradicts x ¯ B ( p ) . We have shown that ¯ B ( p ) B m ( p ) . Let x B m ( p ) , y X ( q 1 , . . . , q m - 1 ) , q m y < 0 . Consider a sequence x n = 1 n y + (1 - 1 n ) x. By convexity, x n * X for any n * IN . Moreover, px n < 0 for all n IN . Show that px ˜ n < 0 for some hyperfinite ˜ n. Suppose that the set A = { n * IN : px n 0 } is non-empty. This set is internal as a definable subset of an internal set * IN (cf. Davis (1977), Theorem 1-8.1). Therefore it has a least element ν * IN \ IN . Take ˜ n = ν - 1 , then x ˜ n x and px ˜ n < 0 , which proves that x ¯ B ( p ) . To complete the proof it suffices to show that m ∈ { 1 , . . . , k } q m X ( q 1 , . . . , q m - 1 ) 0 implies ¯ B ( p ) = B k +1 ( p ) . If x B k +1 ( p ) then px = 0 , so x ¯ B ( p ) . The proof of the inclusion ¯ B ( p ) B k +1 ( p ) goes along the same lines as in the case m k.
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  • Spring '16
  • Equilibrium, Economic equilibrium, General equilibrium theory, Non-standard analysis, Florig

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